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Reference Guide & Formula Sheet for Physics Dr. Mitchell A. Hoselton

Physics − Douglas C. Giancoli

Chapter 01. – Units, Unit Conversion, Symbols Units are important. You will learn the names of lots of units this year. Each unit has a standard one or twoletter abbreviation. You must learn the abbreviations and use them. Start with these few. (The unit of time is the second = s)

Page 1 of 16

Chapter 01. – continued Error and Precision are not the same thing. Error tells us how far the measurement is from the true answer. We will usually report error as Percent Error.

%Error=100%×|Measured− −True|/True

(The unit of distance or displacement is the meter = m) (The unit of velocity is meters per second = m/s) (The unit of acceleration is meters per second squared = m/s2)

Symbols are important. They stand in for known and unknown quantities in the equations you will derive to solve Physics problems. Many of these symbols are considered so standard that everyone everywhere uses the same symbol. This actually simplifies matters is many cases. We will use the standard symbols whenever possible. You will have to learn these as we move along. Start with these few. (The symbol of time is t)

Precision tells us only how consistently a given measuring device can measure values ACCORDING TO ITS MANUFACTURER’S SPECIFICATION. Precision is a measure of the reproducibility and consistency of the results. The measurements can be very consistent and still be consistently wrong, however. Typically, the specification for the precision of a device might be reported something like one of the following, ±0.002 ±1% 3½ digits

(The symbol of distance is d) (The symbol of speed is v or s) (The symbol of displacement is d or r or s) (The symbol of velocity is v) (The symbol of acceleration is a)

Subscripts are important. Subscripts add essential information that must be taken into account. For example, v0 usually indicates the velocity at time zero, while vi and vf usually indicate the initial and final velocities, and vAVE is the average velocity. These are all different velocities. Always read the subscripts. Time is important. Time started 14 billion years ago when the universe appeared. We will not be studying processes that began at the beginning of time. All times that we measure are therefore time intervals, or time differences, if you like. For us time zero always means that time when the clock started. And time t always means the time interval since the clock started. In light of this fact, is it never wrong to replace t with a ∆t in any equation. If the time interval does not start at time zero on the clock, then the time must be written as ∆t. Standard Units: The standard units we use are known as SI units. For now, learn these first few. Measure of length Measure of area Measure of volume Measure of time Measure of velocity Measure of acceleration

meters = m meters2 = m2 = m×m meters3 = m3 = m×m×m seconds = s meters per second = m/s meters per second2 = m/s2

Fine precision is no guarantee of high accuracy, however. Usually the two go together, but sometimes, probably by mistake, they do not. (Hubble Telescope!) Unit Conversion Factors – Remember that all unit conversion factors only change the numerical answer because they change the units in which it is reported. These defined relationships always have very high accuracy, and practically an unlimited number of significant digits (even if the terminal zeroes are not written out).

100 cm = 1 meter Example: Suppose you want to convert 35.0 miles per hour to meters per second. You would need conversion factors based on the following equalities. 1 mile = 5,280 feet 1 foot = 12 inches 1 inch = 2.54 centimeters 100 centimeters = 1 meter 1 hour = 60 minutes 1 minute = 60 seconds

35.0

mi 5280 ft 12 in 2.54 cm 1m × × × × hr 1 mi 1 ft 1 in 100cm 1hr 1 min × × = 15.7 m s 60 min 60s

From each equality, choose the ratio that eliminates an unwanted unit and adds a unit that moves the answer in the desired direction.

Version 6/5/2006

Reference Guide & Formula Sheet for Physics Dr. Mitchell A. Hoselton

Physics − Douglas C. Giancoli

Page 2 of 16

Chapter 02. – Motion Along one Axis Physical quantities for which the direction of their motion or action is an important characteristic must be treated mathematically using vectors, not simple numerical values. Quantities that do not require directional information are called scalars. We begin our study of vectors by studying motion in one direction. This type of vector behaves much like a scalar quantity; only the notation is a little different at this point.

Chapter 02. – continued With the definition of distance in hand we can define the scalar quantity known at the average speed.

Vectors are symbolized with very bold letters. The most important quantities in this section are the instantaneous quantities listed here.

often writing simply as t, but this is only true if the time interval begins at the moment when the clock reads zero. (Think of the clock as a stopwatch.)

Instantaneous position:

x

Instantaneous velocity at time zero: Instantaneous acceleration:

Where ∆t is the time interval between the moment when the object was at the initial position and the moment when it was at the final position. The time interval is

With the definition of the displacement in hand we can define the vector quantity known as the average velocity

Instantaneous position at time zero: x0 Instantaneous velocity:

vAVG = average speed = distance/∆time = d/∆t

vAVG = average velocity = displacement/∆∆time = d/∆ ∆t

v v0 a

(for now, acceleration is assumed to be constant.)

Before we can rigorously define what we mean by instantaneous, we need to define some simpler quantities. The first of these are distance and displacement. In one dimension these might have the same numeric value. d = distance = odometer reading If the object starts at x0 and moves back and forth before settling at its final position, x, then the distance could be much longer than the shortest path between the starting and ending points. On the other hand, the minimum distance is closely related to the displacement.

x| = |d| dMIN = |x x − x0| = |x x0 − x The minimum distance does not include information about the direction of travel; that is the meaning of those absolute value markers. The starting and ending

Instantaneous Position – x – the position of a moving object at one moment in time; also known as an instant of time. Position is always assumed to be instantaneous. Instantaneous Velocity – v – is average velocity over an infinitesimal displacement in an infinitesimal time interval. It is the velocity at one instant. As a practical matter it is usually good enough to measure the average velocity over a short displacement in a brief time interval and then take the ratio of those two measurements to estimate the instantaneous velocity. Constant-Acceleration Linear Motion Once our final definitions for instantaneous position and instantaneous velocity are completed, the following equations of motion apply to all systems that have a constant acceleration. (When the acceleration is constant, the instantaneous and average accelerations have the same magnitude and direction.)

v = v0 + a•t no x (x x − x0) = v0•t + ½•a a•t² v² = v0² + 2•a•(x x − x0) (x x − x0) = ½•(v v0 + v)•t (x x − x0) = v•t − ½•a a•t²

positions are x0 and x, but the scalar quantity called “minimum distance” does not care which is which. Subtraction in either order is permitted.

d = displacement = x − x0 = d The vector quantity called “displacement”, on the other hand, must be calculated as the final position vector minus the initial position vector. That result always gives us the minimum distance and the direction of the motion. For motion along the x-axis, as one example, a positive displacement indicates motion to the right. A negative displacement indicates motion to the left.

no v no t no a no vο

Average velocity can be obtained from the initial and final instantaneous velocities, if and only if the acceleration is constant.

Version 6/5/2006

v

AVE

=

v+v = average velocity 2 0

Reference Guide & Formula Sheet for Physics Dr. Mitchell A. Hoselton

Physics − Douglas C. Giancoli

Chapter 02. – continued Constant Acceleration is rare in nature but common in the problems we will be working. Constant acceleration gives the equations of motion their simplest form and makes them easier to solve. Gravity provides a ready source of objects moving with constant acceleration. Strictly speaking, we cannot define average acceleration until we have a definition for instantaneous velocity. Then the average acceleration is

aAVG = average acceleration = velocity change/∆time = ∆v v/∆t When the acceleration is constant, the instantaneous and average acceleration have the same magnitude and direction. Since the instantaneous acceleration is the same at all moments, the average acceleration must have the same magnitude and direction. One dimensional vectors To this point, we’ve used only vectors that behave exactly like signed numerical values, where the sign indicates the direction along the axis of motion.

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Chapter 03. – continued V + w = (22.8− −20.3) i + (25.3+47.9) j m/s V + w = 2.5 i + 73.2 j m/s = w + V V − w = (22.8−(− −(− −(−20.3)) i + (25.3−47.9) j m/s = 43.1 i − 22.6 j m/s

w − V = (−−20.3−−22.8) i + (47.9−25.3) j m/s = −43.1 i + 22.6 j m/s V − w and w − V, point in opposite directions and both are perpendicular to V + w = w + V V. Projectile Motion – working with components

x− −xο = vx•t y− −yο = vy0•t − ½•g•t²

Horizontal position: Vertical position: Horizontal velocity: Vertical velocity:

vx = v0 cos θ vy = v0 sin θ – gt2

Horizontal acceleration: Vectors can also be thought of as arrows with pointed ends showing the direction of the motion. We could even use these arrows to describe the one-dimensional vectors discussed in this chapter. In the next chapter, where objects are free to move in two dimensions, we will use the arrow representation first. There is also a method that allows us to reuse the vector concepts from this chapter; the signed numbers. We will separate the vectors into what are call their components. Components are independent onedimensional sub-sets of the motion.

Chapter 04. – Newton’s First Law – Law of Inertia. Forces make objects move. No force means no change in the motion. Newton's Second Law – Forces cause acceleration.

Fnet = ΣFExt = msys•asys Newton’s Third Law – Forces are created in pairs. Weight = W

Chapter 03. – Components of a Vector and Vector Addition V = v ∠θ = 34.0 m/s ∠48.0° vx = v cos θ = 34 m/s•(cos 48°) = 22.8 m/s vy = v sin θ = 34 m/s•(sin 48°) = 25.3 m/s

= m•g g

g = 9.80m/s² near the surface of the Earth = 9.795 m/s² in Fort Worth, TX Friction Force = FF

V = vx i + vy j = 22.8 i + 25.3 j m/s

= µ•F FN

If the object is not moving, you are dealing with static friction and it can have any value from zero up to µS FN

W = w ∠θ = 52.0 m/s ∠113.0° wx = w cos θ = 52 m/s•(cos 113°) = −20.3 m/s wy = w sin θ = 52 m/s•(sin 113°) = 47.9 m/s

w = wx i + wy j = −20.3 i + 47.9 j m/s

Vertical acceleration:

ax = 0 ay = −g = constant

If the object is sliding, then you are dealing with kinetic friction and it will be constant and equal to µK FN

Free-Body Diagram – Show all the forces acting on and object. Components must be dashed to distinguish them from forces. You cannot use a force and its components in the same problem.

Version 6/5/2006

Reference Guide & Formula Sheet for Physics Physics − Douglas C. Giancoli

Dr. Mitchell A. Hoselton

Chapter 05. – Uniform Circular Motion - Centripetal Acceleration 2

v a = r

Chapter 06 – continued Mechanical Energy – The Work-Energy Theorem The net work done on a body equals the change in the kinetic energy of the body.

R

Wnet = ∆KE = KEf − KEi = ½•m•vf² − ½•m•vi²

Uniform Circular Motion – Period, Frequency and V

T=

1 1 2πr and f = and v = f T T

Centripetal Force

Page 4 of 16

Mechanical Energy – Gravitational Potential Energy

PEGrav = PEg = m•g•h = m•g•y Hooke's Law – a non-constant force

2

mv = mω r F = r

F = −k•x

2

C

x = displacement from equilibrium k = the spring constant = proportionality constant between the restoring force and the displacement.

Minimum Speed at the top of a Vertical Loop

v = rg

Potential Energy of a spring – Conservative Force Circular Unbanked Track – Car Rounding a Curve. 2

F = µmg = C

mv = ma r

average power = P =

v2 = r•g•tan θ, without friction

momentum = p

mm F =G 12 2 r N•m² / kg² =

Work Fd = = Fv ∆time ∆t

Chapter 07. – Linear Momentum

Universal Gravitation – Conservative Force

where G = 6.67 ×10

= ½•k•x²

Power = rate of work done, unit = J/s = W = watts

R

Banked Circular Track

−11

Work done on a spring = PE = W

= m•v = mass • velocity

Newton's Second Law

6.67 E−11 N•m² / kg²

F = ΣF = net

Ext

∆p mv − mv m (v − v ) = ma = = ∆t ∆t ∆t 0

0

Chapter 06. – Work done by a constant force = F•D •cos

θ

Impulse = Change in Momentum

Where D is the displacement of the mass and θ is the angle between F and D. unit : N•m = J Work done by a varying force On a graph of Force vs displacement the work is the area between the curve and the x-axis. Later on we will evaluate this are by taking the anti-derivative of the function that describes the force in terms of position.

F•∆ ∆t = ∆ ∆p = ∆( ∆(m•v) Conservation of Momentum in Collisions

mA•vA + mB•vB = mA•vA’ + mB•vB’ Sum of Momenta before the Collision

Sum of Momenta after the Collision

Center of Mass – point masses on a line (x only), on a plane (x and y only), or filling space (x, y, and z)

Mechanical Energy –Kinetic Energy

KELinear = K = ½•m•v² Version 6/5/2006

xcm = Σ( Σ(mi•xi) / Mtotal ycm = Σ( Σ(mi•yi) / Mtotal zcm = Σ( Σ(mi•zi) / Mtotal

Reference Guide & Formula Sheet for Physics Physics − Douglas C. Giancoli

Dr. Mitchell A. Hoselton Chapter 08. – Angular Distance – Radian measure

θ = arc length/radius = l/r = s/r 360º = 2π π radians

Chapter 09. – Elasticity; Stress and Strain (Assumes objects stretch according to Hooke’s Law as long as they are not stretched passed the proportional limit.)

Angular Speed vs. Linear Speed Linear speed = v

= r•ω ω = radius • angular speed

For tensile stress

F = k•∆ ∆L = (EA/L0)•∆ ∆L

Constant Angular-Acceleration in Circular Motion ω = ωο + α•t no θ θ−θο = ωο•t + ½•α•t² no ω 2

2

ω = ωο + 2•α•((θ−θο) θ−θο = ½•(ωο + ω)•t θ−θο = ω•t - ½•α•t² Torque =

F = applied force E = elastic (or Young’s) modulus A = cross-sectional area – perpendicular to the force L0 = the original length ∆L = change in the length

no t no α no ωο

∆L/L0 = (1/E)•(F/A) strain = (1/E)•(stress) E = (stress / strain)

τ = F•L L•sin θ

Where θ is the angle between F and L; unit: N•m Newton's Second Law for Rotation torque = τ

= I•α α moment of inertia = ICM = m•r² (for a point mass)

Compressive stress is the exact opposite of tensile stress. Objects are compressed rather than stretched. As for springs the equations are the same for both tensile and compressive stress and the same elastic modulus is used for both calculations.

Rotational Kinetic Energy (See LEM on last page)

∆L/L0 = −(1/E)•(F/A) strain = − (1/E)•(stress) E = − (stress / strain)

2

KErotational = ½•I•ω ω = ½•I• (v / r)2 2 KErolling w/o slipping = ½•m•v2 + ½•I•ω ω Moment of Inertia – ICM point mass cylindrical hoop solid cylinder or disk solid sphere hollow sphere

ICM = m•r2 ICM = m•r2 ICM = ½ m•r2 ICM = 2/5 m•r2 ICM = ⅔ m•r2 ICM = 1/12m•L2

There is a negative sign because the length decreases as the force increases (∆ ∆L is negative). Shear stress is the application of two forces that distort an object (like deforming a rectangle into a parallelogram). The forces are equal and opposite (parallel, but not oriented to directly oppose each other). (A second pair of matched forces is also required to maintain equilibrium while the stress is applied.)

thin rod (center) When the thin rod is rotated about its end rather than about is center of mass, the moment of inertia becomes thin rod (end) Angular Momentum =

Page 5 of 16

IEnd = ⅓ m•L2 L = I•ω = m•v•r•sin θ

Angular Impulse equals CHANGE IN Angular Momentum

∆ ∆L = τAverage•∆ ∆t = ∆(I•ω ω)

Version 6/5/2006

∆L /L0 = (1/G)•(F/A) strain = (1/G)•(stress) G = (stress / strain) G = shear modulus A = area – Parallel to the force. L0 = original length of object ∆L = change in length due to force Note that ∆L is perpendicular to L0.

Reference Guide & Formula Sheet for Physics Dr. Mitchell A. Hoselton

Physics − Douglas C. Giancoli

Chapter 09. – continued Bulk Stress - When the force is applied uniformly all over an object, we expect its volume to shrink. When applied this way, the ratio of force to area is called “pressure”, and ∆P is the change in pressure that induces a corresponding change in the volume.

Chapter 10. – continued Poiseuille's Equation (Laminar flow in horizontal tubular pipes.) Q = volume flow rate of fluid = m3/s 4 Q = (π π•r ) ∆P / (8•η η•L)

r = inside radius of pipe = m ∆P = P1 – P2 = Pressure change = Pa η = coefficient of viscosity = Pa•s L = length of pipe = m

∆V/V0 = −(1/B)•(∆P) strain = −(1/B)•(stress) B = −(stress / strain) B = bulk modulus V0 = initial volume of the material ∆V = Change in the volume ∆P = change in pressure The minus sign indicates that the volume decreases when the pressure increases. When ∆P is positive, ∆V is negative, and vice versa. One of the two is always negative.

Page 6 of 16

Bernoulli's Equation

P + ρ•g•h + ½•ρ•v ² = constant QVolume Flow Rate = A1•v1 = A2•v2 = constant Chapter 11. – Period of Simple Harmonic Motion – Ideal Spring

m k

T = 2π

Chapter 10. – Pressure under Water (or immersed in any liquid)

P=ρ ρ•g•h

also f = 1/ T

where k = spring constant, and m is the mass. Simple Pendulum

P = Pressure at depth h = depth below the surface ρ = density of the fluid

T = 2π

L g

also f = 1/ T

Density = mass / volume

(

m ρ = unit : kg / m3 V

where L is the length of the pendulum and g is the local acceleration due to gravity.

)

Velocity of Periodic Waves

f=1/T v = f •λ λ = λ/Τ

Buoyant Force - Buoyancy

FB = ρ•V•g = mDisplaced fluid•g = we...